Mathematical Sciences: Some Problems in Harmonic Analysis Related to Wavelets

这项数学研究的主旨是围绕小波理论的最新发展的应用。第一个应用将集中在由Calderon-Zygmund型算子生成的线性算子的代数上。这些算子出现在偏微分方程组的分析中。运算符太复杂，不能单独研究；当前研究的目标是选择小波基，其中的运算符可以表示为几乎对角矩阵。从这个表示中，人们可以读出运算符的许多隐藏属性。第二个研究方向是高维空间中的小波分解。这项工作将从用某种程度上尊重对称性的小波来表示径向函数开始。根本的困难在于，人们不能对称化一维小波并保持所需的平移性质。在发展一种保留了许多理想性质的新的小波型函数方面已经取得了一些进展。本研究的第三个方向是使用小波型表示来区分不同Riesz变换在函数空间上的行为。小波理论为谐波分析领域引入了一种非常新的工具。它包含了用于表示函数的高效理论方法的元素，以及用于计算在时间和频率上同时处于局部的函数的各种展开的自然算法。这些应用在科学和工程界才刚刚开始感受到。

The thrust of this mathematical research revolves around applications of recent developments in wavelet theory. The first application will focus on algebras of linear operators generated by those of Calderon-Zygmund type. These operators arise in the analysis of partial differential equations. The operators are too complex to study individually; the goal of current research is to choose wavelet bases in which the operators may be represented as almost-diagonal matrices. From this representation, one can read off many hidden properties of the operators. A second line of investigation concerns wavelet decompositions in higher dimensional space. This work will begin with the representation of radial functions in term of wavelets which somehow respect the symmetry. The fundamental difficulty is that one cannot symmetrize one-dimensional wavelets and maintain the desired translation properties. Some progress has been made in developing a new wavelet-type function with many desirable properties preserved. A third direction of this research is that of distinguishing the behavior of the different Riesz transforms on functions spaces using wavelet-type representations. Wavelet theory has introduced an extraordinary new tool into the field of harmonic analysis. It contains the elements of a highly effective theoretical method for the representation of functions along with natural algorithms for the computation of various expansions of functions which are simultaneously local in time and frequency. There applications are only beginning to be felt in the scientific and engineering community.